Method and system for performing signal analysis using a hybrid photonic-electronic processor

ABSTRACT

An apparatus for generating a time-delayed product of two independent signals includes a fixed-wavelength laser. A first optical modulator is optically coupled to the fixed-wavelength laser and configured to modulate a fixed wavelength optical carrier with a first input signal of a set of input signals. The apparatus also includes a tunable laser. A second optical modulator is optically coupled to the tunable laser and configured to modulate a tunable optical carrier with a second input signal of the set of input signals. The apparatus also includes a dispersive element coupled to the second optical modulator, a first optical detector coupled to the dispersive element, a third optical modulator optically coupled to the first optical detector and the first optical modulator, an optical 90-degree hybrid element optically coupled to the third optical modulator, and a plurality of optical detectors optically coupled to the optical 90-degree hybrid element.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional Application No.63/154,506, filed Feb. 26, 2021, the contents of which are herebyincorporated by reference in their entirety for all purposes.

The following regular U.S. patent applications (including this one) arebeing filed concurrently, and the entire disclosure of the otherapplication is incorporated by reference into this application for allpurposes:

-   -   Application Ser. No. ______, filed Feb. 25, 2022, entitled        “METHOD AND SYSTEM FOR PERFORMING SIGNAL ANALYSIS USING A HYBRID        PHOTONIC-ELECTRONIC PROCESSOR;” and    -   Application Ser. No., filed Feb. 25, 2022, entitled “METHOD AND        SYSTEM FOR PERFORMING SIGNAL ANALYSIS USING A CORRELATIVE        RECEIVER.”

BACKGROUND OF THE INVENTION

Real-time signal analysis is strictly dictated by the waveformbandwidth. Current electronic processors allow for complex analysis forsignals with bandwidth smaller than 1 GHz. Examples of currently viablereal-time electronic processing include Fourier and Hilbert transforms,correlation, convolution, pattern matching, and cyclostationaryanalysis. To perform any of these, a physical signal must first bedigitized, mandating the use of an analog-to-digital (ADC) convertorprocessor in the front-end. While a number of applications can be servedby sub-GHz signal analysis, a number of important applications requirereal-time analysis of signals with bandwidth exceeding 1 GHz.

With the emergence of high-capacity wireless communications, spectralanalysis of wideband radio-frequency signals, associated with militaryapplications in the past, has emerged as one of the most importantapplications. Current 5G wireless infrastructure incorporates signalingin excess of 1 GHz, requiring real-time spectral analysis for thepurposes of provisioning, network surveillance, security and trafficconflict resolution. In electronic warfare, higher-order spectralanalysis of the contiguous radio-frequency (RF) band (e.g., 1 kHz-110GHz) is dictated by a need to identify, intercept and geolocate anyemitter and classify its operation. In light wave (e.g., fiber optic)communication, it is necessary to analyze channels with aggregatecapacity exceeding 400 Gbit/sec in order to provide real-time securityand reverse transmission penalties. Current electronic processorspossess neither the precision nor speed to satisfy these applications,causing a need for improved methods and systems that can addressanalysis of wideband signals.

SUMMARY OF THE INVENTION

In accordance with various aspects of the present disclosure,apparatuses and methods for wideband signal analysis using hybridphotonic-electronic processing are provided.

According to various aspects, there is provided an apparatus forgenerating a time-delayed product of two independent signals.

One embodiment includes an apparatus comprising: a fixed-wavelengthlaser operable to provide a fixed wavelength optical carrier; a firstoptical modulator optically coupled to the fixed-wavelength laser andconfigured to modulate the fixed wavelength optical carrier with a firstinput signal of a set of input signals; a tunable laser operable toprovide a tunable optical carrier; a second optical modulator opticallycoupled to the tunable laser and configured to modulate the tunableoptical carrier with a second input signal of the set of input signals;a dispersive element coupled to the second optical modulator; a firstoptical detector coupled to the dispersive element; a third opticalmodulator optically coupled to the first optical detector and the firstoptical modulator; an optical 90-degree hybrid element optically coupledto the third optical modulator; and a plurality of optical detectorsoptically coupled to the optical 90-degree hybrid element and configuredto convert outputs of the optical 90-degree hybrid element intoelectrical signals.

Another embodiment includes an apparatus comprising: an optical sourceoperable to provide a frequency comb including a plurality of opticalcarriers; a first optical modulator optically coupled to the opticalsource and electrically coupled to a first antenna operable to receive afirst input signal of a set of input signals; a tunable laser operableto provide a tunable optical carrier; a second optical modulatoroptically coupled to the tunable laser and electrically coupled to asecond antenna operable to receive a second input signal of the set ofinput signals; a dispersive element optically coupled to the secondoptical modulator; a first optical detector optically coupled to thedispersive element; a third optical modulator optically coupled to thefirst optical modulator and electrically coupled to the first opticaldetector; a wavelength-division demultiplexer optically coupled to thethird optical modulator and the optical source; a plurality of optical90-degree hybrid elements, each of the plurality of optical 90-degreehybrid elements being coupled to the wavelength-division demultiplexer;a plurality of optical detectors, each of the plurality of opticaldetectors being coupled to an output of one of the plurality of optical90-degree hybrid elements; and a plurality of analog-to-digitalconverters, each of the plurality of analog-to-digital converters beingelectrically coupled to one of the plurality of optical detectors.

Numerous benefits are achieved by way of the present invention overconventional techniques. For example, embodiments of the presentinvention provide methods and systems for real-time processing ofradiofrequency (RF) input signals which may have bandwidth larger thanseveral hundred GHz. These and other embodiments of the invention, alongwith many of its advantages and features, are described in more detailin conjunction with the text below and attached figures

BRIEF DESCRIPTION OF THE DRAWINGS

Aspects of the present disclosure now will be described more fullyhereinafter with reference to the accompanying drawings, which areintended to be read in conjunction with both this summary, the detaileddescription and any preferred and/or particular embodiments specificallydiscussed or otherwise disclosed. The various aspects may, however, beembodied in many different forms and should not be construed as limitedto the embodiments as set forth herein; rather, these embodiments areprovided by way of illustration only and so that this disclosure will bethorough, complete and will fully convey the full scope to those skilledin the art.

FIG. 1 is a block diagram of a hybrid photonic-electronic processoraccording to an embodiment of the present invention.

FIG. 2A is a set of plots illustrating a first case, where thebandwidths of the detector and of the ADCs are sufficiently largeaccording to a first embodiment of the present invention.

FIG. 2B is a set of plots illustrating a second case, where thebandwidths of the detector and of the ADCs are not sufficiently largeaccording to a second embodiment of the present invention.

FIG. 3 is a block diagram of an implementation of the delay arm usingcoherent detection and modulation according to an embodiment of thepresent invention.

FIG. 4 is a block diagram of an implementation of the delay arm in anincoherent form according to an embodiment of the present invention.

FIG. 5 is a plot illustrating the delayed product of the two RF inputssampled by the ADC.

FIG. 6 is a cross-ambiguity function (CAF) realized as a table.

FIG. 7A is a plot of a CAF table generated by maintaining a constantdelay τ and varying the cyclical frequency α.

FIG. 7B is a plot of a CAF table generated by varying the delay τ andmaintaining a cyclical frequency α.

FIG. 8 is a simplified schematic diagram illustrating a hybridphotonic-electronic processor utilizing a parallelized acquisitionsystem according to an embodiment of the present invention.

FIG. 9 is a plot showing CAF tables generated using a parallelizedhybrid photonic-electronic processor

FIG. 10 is a plot showing the relation between the maximal delay range(τ_(RANGE)) of a tunable laser, the time utilized to sweep the laserwavelength over the delay range (t_(Sweep)), and the time to acquire asingle delay state (t_(Dwell)).

FIG. 11 is a plot showing the delay variation during acquisition time.

FIG. 12 is a plot illustrating constraints defined by the tunable lasersweep rate.

FIG. 13 is a plot illustrating two acquisition conditions in which thesweep time of the tunable laser is constant and the measurement periodsdiffer.

FIG. 14 is a simplified flowchart illustrating a method of generating across-ambiguity function (CAF) table according to an embodiment of thepresent invention.

DETAILED DESCRIPTION OF THE INVENTION

The present invention relates generally to wideband signal analysis witha hybrid photonic-electronic processor. More particularly, embodimentsof the present invention provide methods and systems that acquire andprocess high bandwidth signals (e.g., larger than 100 GHz) in real-time.However, embodiments of the present invention are applicable to avariety of wideband signal analysis systems.

FIG. 1 is a block diagram of a hybrid photonic-electronic processor 100according to an embodiment of the present invention. The hybridphotonic-electronic processor 100 includes a front end dedicated toacquiring two arbitrary RF signals (e.g., a first RF signal y(t) and asecond RF signal x(t)) and performing its operation in the photonicdomain before converting the result to digital form and completing theprocessing chain.

The hybrid photonic-electronic processor 100 may receive radiofrequency(RF) input (e.g., the arbitrary signals x(t) and y(t)) through a firstantenna 102 and a second antenna 104. As an example, the first antenna102 and the second antenna 104 can receive RF signals having a bandwidthup to hundreds of GHz. The hybrid photonic-electronic processor 100converts the inputs received by the first antenna 102 and second antenna104 from the RF domain to the optical domain via optoelectronicmodulators.

The hybrid photonic-electronic processor 100 includes a firstoptoelectronic modulator 106, a second optoelectronic modulator 108, anda third optoelectronic modulator 118. The hybrid photonic-electronicprocessor 100 may further include a tunable laser 110. The output of thetunable laser 110 (e.g., an optical carrier) is provided to the secondoptoelectronic modulator 108 using a suitable optical connection. Asillustrated in FIG. 1, the tunable laser 110 is capable of tuning itscenter frequency over a frequency range Δf that ranges from f₁ to f₂.The RF input y(t) of the second antenna 104 modulates the opticalcarrier generated by the tunable laser 110 via the second optoelectronicmodulator 108, converting the RF input y(t) from the RF domain to theoptical domain.

Referring to FIG. 1, a delay τ in the first RF signal y(t) is generatedby transmitting the modulated optical carrier produced at the output ofthe second optoelectronic modulator 108 over a second dispersive line114. The relative delay τ between modulated carriers, centered at f₁ andf₂ and separated by Δf, is defined by τ=D×L×c×Δf/f², where D is thedispersion of the second dispersive line 114, L is the length of thesecond dispersive line 114, c is the speed of light, and f is thefrequency. As an example, for a second dispersive line 114 with L=1 kmlong, conventional dispersion compensating fiber that possesses adispersion D=5×10⁻⁴ s/m² will induce a delay of τ=5 ns when a tunablelaser operating at a wavelength of 1550 nm (i.e., a frequency of f=193THz) is tuned over a frequency range of Δf=1250 GHz (i.e., 10 nm). Thepractical delay range of a processor using a fiber as a dispersion linecan be limited by the fiber loss α=0.25 dB/km and the tunable range ofthe laser carrier. An acceptable loss (e.g., around 10 dB) and a commontunable range of a 1550 nm laser (i.e. approximately 100 nm) result in adelay range of 2000 ns being achieved.

The delayed signal output (labeled as y(t−τ) in FIG. 1) of the seconddispersive line 114 is then sent to a detector element 116, which can bereferred to as a first optical detector. The detector element 116 willconvert the delayed signal from the optical domain back to the RFdomain. Thus, the embodiment illustrated in FIG. 1 utilizes the tunablelaser 110, the second optoelectronic modulator 108, the seconddispersive line 114, and the detector element 116 (e.g., an intensitydetector) to implement a tunable delay line that delays the arbitraryfirst RF input y(t) to produce delayed signal y(t−τ) used to drive thethird optoelectronic modulator 118.

The hybrid photonic-electronic processor 100 may further comprise afixed-frequency laser 120. The output of the fixed-frequency laser 120(e.g., a fixed frequency optical carrier) may be split into twoportions, with an upper portion as illustrated in FIG. 1 being providedto the first optoelectronic modulator 106 using a suitable opticalconnection and a lower portion as illustrated in FIG. 1 being providedto a frequency shifter 150. The second RF input x(t) received at thesecond antenna 104 may be converted from the RF domain to the opticaldomain by modulating the output of the fixed-frequency laser 120 via thefirst optoelectronic modulator 106. As a result, the optical signaloutput by the first optoelectronic modulator 106 carries the second RFinput signal x(t). In some embodiments, first optoelectronic modulator106 is implemented as a dual-port Mach-Zehnder modulator and, in otherembodiments, the first optoelectronic modulator 106 is implemented as adual-port, nested Mach-Zehnder device.

The modulated signal produced by the first optoelectronic modulator 106is transmitted to the third optoelectronic modulator 118 via the firstdispersive line 112. The frequency shifter 150 may be configured toshift the frequency of the fixed-frequency laser 120 by an arbitraryamount. This will act as a local oscillator for the system, and beutilized during detection of the product that will be generated in theupper portion.

In the embodiment illustrated in FIG. 1, the first dispersive line 112introduces a fixed delay to the optical signal that is received at thethird optoelectronic modulator 118. The description provided hereinassumes that the first dispersive line 112 is not utilized and that thesecond dispersive line 114 is designed and implemented in the absence ofthe first dispersive line 112. Thus, the first dispersive line 112 isnot required to implement the processor functionality described hereinand is inserted as an option to shift the delay center (e.g., a centerfrequency around which the hybrid photonic-electronic processor 100 willacquire the signal) associated with the second dispersive line 114, thetuning range of the tunable laser 110, and the first dispersive line112. Accordingly, use of first dispersive line 112 can enable anincreased delay. One of ordinary skill in the art would recognize manyvariations, modifications, and alternatives.

At the third optoelectronic modulator 118, the output of the detectorelement 116 (RF signal y(t−τ)) is used as an RF drive input to modulatethe second RF input x(t) received at the second antenna 104, resultingin the generation of the multiplication state x(t)×y(t−τ) in the opticaldomain and residing on an optical carrier at the output of the thirdoptoelectronic modulator 118, which is subsequently filtered using thefiltering element 122. Thus, using inputs of the delayed first RF inputssignal y(t−τ) in the optical domain and the second RF input signal x(t)in the optical domain, which is potentially delayed using the firstdispersive line 112, the third optoelectronic modulator 118 produces anoutput equal to the product of x(t)×y(t−τ). As described more fullybelow, this product is the kernel of the cross-correlation function.

Accordingly, using the hybrid photonic-electronic processor 100illustrated in FIG. 1, a multiplication state can be generated betweentwo arbitrary RF signals, namely the RF inputs y(t) received at thefirst antenna 102 and x(t) received at the second antenna 104, and foran arbitrary delay τ between the two arbitrary RF signals, given thatthe delay τ falls within the ranges defined by the dispersion of thesecond dispersive line 114 and the tuning range of the tunable laser 110described above. Specifically, this means that the kernel of thecross-correlation function can be generated at the output of the thirdoptoelectronic modulator 118 in a computation-free manner as shown inequation (1):

C(T)=∫_(−τ) ^(τ) x(t)×y(t−τ)dt  (1)

where −τ<t<τ is the integration interval used to estimate thecross-correlation function. The filtering element 122 can be used tocomplete the integration in the [−τ, τ] bounds by designing a matchingfiltering function. The filtering element 122 can be eliminated if thebackplane digital processer 130 possesses sufficient memory to completethe integration step. If the output of the hybrid photonic-electronicprocessor 100 is intended to generate either the product or the delayedproduct of the inputs received at the first antenna 102 and the secondantenna 104, the integration done by the filtering element 122 and/orbackplane digital processor 130 may not be utilized. In someembodiments, the backplane digital processor 130 performs a discreteFourier transform on the output of the ADC elements 128 and 129.

As illustrated in FIG. 1, a coherent detection implementation is used inwhich a tunable frequency shift is applied by the frequency shifter 150to the optical carrier generated by fixed-frequency laser 120. Thisfrequency shifted optical carrier is utilized as a local oscillator andcombined in a coherent receiver to detect the multiplication statex(t)×y(t−τ).

The output of the third optoelectronic modulator 118 is combined withthe optical carrier generated by the fixed-frequency laser 120 and sentto an optical 90-degree hybrid element 124. The optical 90-degree hybridelement 124 is configured to separate quadratures (e.g., an orthogonaldecomposition may be performed by the optical 90-degree hybrid element124 to separate any signal into orthogonal components known asquadratures; an example of such a method is separating the real andimaginary components in base band signals) of the output (optionallyfiltered using filtering element 122) of the third optoelectronicmodulator 118. The output of the optical 90-degree hybrid element 124 isprovided to the set of balanced detectors 126 and 127, also referred toas balanced receivers, in order to perform coherentoptical-to-electrical conversion. The output of each balanced detectoris subsequently digitized by a set of analog-to-digital conversion (ADC)elements 128 and 129 and provided to the backplane digital processor130. Thus, the hybrid photonic-electronic processor 100 illustrated inFIG. 1 utilizes a direct modulation and direct detection methodology tocompute F(x(t)×y(t−τ)) as described more fully herein. As describedherein, the signals output by the by the ADCs are combined inside theprocessor 130 as a complex number, the real part from one ADC and theimaginary part from the other ADC.

FIGS. 2A and 2B illustrate two different conditions defined by therelationship between the detector bandwidth and the bandwidth of the setof ADCs used in the optical-to-electrical conversion stage.

FIG. 2A is a set of plots illustrating a first case, where the bandwidthof the detector and the bandwidth of the ADCs are sufficiently largeaccording to a first embodiment of the present invention. FIG. 2A showsa case where the bandwidth of the detector and the bandwidth Δf_(ADC)204 of the ADC elements 128 and 129 are at least greater than thebandwidth Δf_(P) 202 of the multiplication state of the two inputsreceived at the first antenna 102 and second antenna 104. The top plotin FIG. 2A is the Fourier transform of the product x(t)×y(t−τ) as afunction of frequency. The bandwidth Δf_(P) can also be referred to asthe bandwidth of the product in the frequency domain.

In this case, the optical carrier generated by fixed-frequency laser120, which is illustrated in the lower plot of FIG. 2A, does not need tobe frequency shifted by the frequency shifter 150 and is combined withthe output of the third optoelectronic modulator 118 as a fixed localoscillator. The bandwidth of the detector and the bandwidth Δf_(ADC) ofthe ADC elements 128 and 129 will typically, at a minimum, be greaterthan the sum of the bandwidths of the two RF inputs y(t) and x(t)received at the first antenna 102 and the second antenna 104. The ADCelements may then be centered to acquire signals at frequency 210 andthe bandwidth Δf_(ADC) 204 will be sufficient to detect the completeFourier transform of the multiplication state F(x(t)×y(t−τ)), which hasa bandwidth Δf_(P) 202.

FIG. 2B is a set of plots illustrating a second case, where thebandwidth of the detector and of the ADCs are not sufficiently largeaccording to a second embodiment of the present invention. FIG. 2B showshow to relax the ADC bandwidth requirements through an example where thedetector and the bandwidth Δf_(ADC) 205 of the ADC elements 128 and 129is less than the bandwidth Δf_(P) 202 of the multiplication state of thetwo inputs received at the first antenna 102 and second antenna 104. Thetop plot in FIG. 2B is the Fourier transform of the product x(t)×y(t−τ)as a function of frequency.

In this case, the optical carrier generated by fixed-frequency laser 120may be frequency shifted by an arbitrary change in frequency Δf 206created by the frequency shifter 150 and combined with the output of thethird optoelectronic modulator 118. This frequency shift Δf 206 changesthe center of the ADCs acquisition position from, for example, frequency210 to frequency 212 as is illustrated in the lower plot of FIG. 2B. Asa result, the ADC elements 128 and 129 then operate over a bandwidthΔf_(ADC) 205 centered on frequency 212. Subsequent steps utilize adifferent frequency shift provided by the frequency shifter 150 toacquire the entire multiplication state bandwidth Δf 202. Since, in thisexample, the bandwidth Δf_(P) 202 of the multiplication state cannot becovered by a single acquisition step of the ADC elements, the frequencyshift Δf 206 corresponding to the frequency of the fixed-frequency laser120 is performed a number of times. Specifically, during eachacquisition step, a portion of F(x(t)×y(t−τ)) is acquired over abandwidth of Δf_(ADC) 205, resulting in the acquisition ofF(x(t)×y(t−τ)) in WA/Arc steps. When the multiplication product is real,its spectrum is symmetric, and so allows the acquisition scan to behalved (i.e., utilizing only Δf_(P)/Δf_(ADC)/2 steps).

The signal delay arm (e.g., the combination of the second optoelectronicmodulator 108, second dispersive line 114, and detector element 116) canbe implemented in both coherent and incoherent forms. This type ofimplementation may be used to reduce distortion at the output, resultingin more accurate calculations of equation (1).

FIG. 3 is a block diagram of an implementation of the signal delay armusing coherent detection and modulation. A coherent modulator term meansthat a complex (I/Q) RF signal is used to drive the optical modulator toencode both phase and amplitude onto the optical carrier. Thus, incontrast with FIG. 1, which illustrated the use of direct modulation anddetection, the embodiment illustrated in FIG. 3 utilizes coherentmodulation and detection. The hybrid photonic-electronic processor 300illustrated in FIG. 3 shares some common elements with the hybridphotonic-electronic processor 100 illustrated in FIG. 1 and thedescription provided in relation to FIG. 1 is applicable to FIG. 3 asappropriate. Differences in the systems are discussed herein.

The output of the tunable laser 310 is split: the upper portion of thelaser output is modulated by the first RF input y(t) received by thefirst antenna 302 and delayed over the second dispersive line 314; thelower portion of the laser output is not modulated and serves as a localoscillator for a coherent detector 316. Coherent detector 316 generatesboth quadrature and phase (I/Q) output in order to drive a thirdcoherent optoelectronic modulator 318. Correspondingly, the firstcoherent optoelectronic modulator 306 is driven by complex (I/Q) RFinput x(t) received at the second antenna 304, allowing the thirdcoherent optoelectronic modulator 318 to generate a complex productx(t)×y(t−τ) at its output. The first dispersive line 112 illustrated inFIG. 1 has been removed, but may be utilized as discussed previously foroptimization.

The output of the third coherent optoelectronic modulator 318 isfiltered using filter 322 and is combined with the optical carriergenerated by the fixed-frequency laser 320 and sent to an optical90-degree hybrid element 324. The output of the optical 90-degree hybridelement 324 is provided to the set of balanced detectors 326 and 327,also referred to as balanced receivers, in order to perform coherentoptical-to-electrical conversion. The output of each balanced detectoris subsequently digitized by a set of analog-to-digital conversion (ADC)elements 328 and 329 and provided to the backplane digital processor330. Accordingly, the embodiment illustrated in FIG. 3 can utilizecoherent techniques to provide an output of processor 330 equal to thecross ambiguity function R_(α) ^(T)(τ).

FIG. 4 is a block diagram of an implementation of the delay arm in anincoherent form. The hybrid photonic-electronic processor 400illustrated in FIG. 4 shares some common elements with the hybridphotonic-electronic processor 100 illustrated in FIG. 1 and thedescription provided in relation to FIG. 1 is applicable to FIG. 4 asappropriate. Differences in the system are discussed herein.

In this implementation, any optoelectronic modulator may be implementedas a single-sideband optical modulator 460. The single-sideband opticalmodulator 460 may be implemented by a combination of an optical90-degree hybrid 462 receiving an RF input and outputting the RF inputalong with a 90-degree shifted copy, sending it to an optical modulator464 which further takes an optical signal as an input to be modulated. Afirst incoherent optoelectronic modulator 408 is driven by the first RFinput y(t) received by the first antenna 402 and its 90-degree shiftedcopy in order to generate a signal-sideband modulated output. In thisconfiguration, the DC component of the modulator output acts as a localoscillator. When received by an incoherent detector element 416, thebeating between the local oscillator and the optical sideband generatesthe original RF signal. In addition to eliminating the need for use ofcoherent modulators (e.g., the first coherent optoelectronic modulator306, the second coherent optoelectronic modulator 308, the thirdcoherent optoelectronic modulator 318 as illustrated in FIG. 3) andcoherent detector (e.g., the coherent detector 316 as illustrated inFIG. 3), single-sideband modulation also reduces or minimizes distortionimposed by the second dispersive line 414. The hybridphotonic-electronic processor 400 illustrated in FIG. 4 does thisreduction in distortion in the front end (e.g., via the first incoherentoptoelectronic modulator 408, and the second incoherent optoelectronicmodulator 406) as opposed to the hybrid photonic-electronic processor300 illustrated in FIG. 3 where it is done via the delay arm (e.g., thesecond coherent optoelectronic modulator 308, the second dispersive line314, and the coherent detector 316).

Embodiments may use a detector and analog-to-digital converter with abandwidth that is lower than the bandwidth of the x(t)y(t−τ) product asillustrated in FIG. 2. The correlation output is generated by assemblingconsecutive signal acquisitions, defined by the tuning sweep time of thetunable laser (e.g., the tunable laser 110). A more general functionshown as equation (2), referred to as the cross-ambiguity function (CAF)R_(α) ^(t) (τ), is often sought in spectral analysis:

R _(α) ^(T)(τ)=∫_(−T/2) ^(T/2) x(t)×y(t−τ)e ^(j2παt) dt  (2)

where α is defined as the cyclical frequency, following the standardterminology from cyclostationary analysis. Note that the conjugation ofone of the inputs leads to the definition shown in equation (3):

R _(α) ^(T)(τ)=∫_(−T/2) ^(T/2) x(t)×y*(t−τ)e ^(j2παt) dt.  (3)

For the purpose of this disclosure, the conjugation of the input signalis assumed as an option, as the front end of the processor can provideboth conjugate and non-conjugate versions of the signal, as describedlater. For the purpose of spectral analysis, it is often useful to workusing the Fourier transform of the CAF, defined as the spectralcorrelation function (SCF). The embodiments illustrated in the figuresand described herein generate a two-dimensional CAF and subsequentlycompute the SCF by a single Fast Fourier Transform (FFT), which iswithin practical limits for off-the-shelf electronic processors.

Referring to the notation seen in FIG. 3, the output of the balancedreceiver 326 is sampled by the ADC 328 that converts the analog waveform(e.g., the delayed product of the first RF input y(t) received by thefirst antenna 302 and the second RF input x(t) received by the secondantenna 304) to a set of sample data shown in FIG. 5.

FIG. 5 is a plot illustrating the delayed product of two RF inputssampled by the ADC, for example ADC elements 128 or 129. The delayedproduct may be sampled by ADC elements of the embodiments describedherein (e.g., the ADC elements 128 and 129 of FIG. 1 or the ADC elements328 and 329 of FIG. 3). The ADC may sample at every t_(sample) seconds,which may then be varied to change the accuracy with respect to the timeof the delayed product measure.

Prior to the analog-to-digital conversion step, the CAF can also beacquired by integrating the delayed product (e.g., by the filteringelement 122 illustrated in FIG. 1 or the backplane digital processor130):

R _(α) ^(T)(τ)=∫_(−T/2) ^(T/2) x(t)×y*(t−τ)e ^(j2παt) dt.  (4)

The equivalent result is achieved without a filter, in the discretedomain by recognizing that:

∫_(−T/2) ^(T/2) x(t)y(t−τ)e ^(j2παt) dt→Σ _(k=1) ^(N) x(kδt)y(δt−τ)e^(−j2παkδt)  (5)

where δt=t_(sample). Similarly, the Fourier transform at the left sideof this relation has a corresponding equivalent in the discrete domain:

$\begin{matrix}{{\int_{- \frac{T}{2}}^{\frac{T}{2}}{{x(t)}{y( {t - \tau} )}e^{{- j}2\pi\alpha t}{dt}}} = { {F( {{x(t)}{y( {t - \tau} )}} )}arrow{\underset{k = 1}{\sum\limits^{N}}{{x( {k\delta t} )}{y( {{k\delta t} - \tau} )}e^{{- j}2{\pi\alpha}k\delta t}}}  = {{DFT}( {{x( {k\delta t} )}{y( {{k\delta t} - \tau} )}} )}}} & (6)\end{matrix}$

where DFT refers to the discrete Fourier transform, ordinarilyimplemented via the Fast Fourier Transform (FFT) algorithm. This meansthat the CAF can be realized in the form of a table.

FIG. 6 is a CAF table 600 realized as a table. As illustrated in FIG. 6,for a selected wavelength of a tunable laser (e.g., the tunable laser110), a specific value of the delay 604 (e.g., τ in FIG. 6 may beequivalent to the delay τ described with respect to FIG. 1) is defined.When the wavelength of the tunable laser is fixed at this selectedwavelength, the delay product is acquired and the CAF value C_(α) ^(T)(τ) for the selected τ is generated at the output of the backplaneprocessor (e.g., the backplane digital processor 130), indicated by thetile labeled as Processed Channel 602 in FIG. 6. The length of the CAFtable segment, the cyclic bandwidth 606, is given by the bandwidth ofthe ADC circuit used to convert the analog delay product to the digitaldomain.

It should be noted that, for purposes of clarity, the delay 604represented by τ is illustrated in FIG. 6 as having a finite dimensionindicated by the parenthesis aligned with the ordinate axis (i.e., they-axis representing the Delay τ in nanoseconds). However, it will beappreciated that this is merely for purposes of illustration and theprocessed channel 602 will be associated with a specific value of thedelay 604 and a range of cycle frequencies given by the bandwidth of theADC circuit used to convert the analog delay product to the digitaldomain. Accordingly, processed channel 602 could also be illustrated inFIG. 6 as a line extending over a range of cycle frequencies. Thus, asdiscussed more fully in relation to FIG. 7B, the delay τ is swept over arange of values by tuning the wavelength of the tunable laser, enablingformation of the CAF table.

FIG. 7A shows a CAF table 700A generated by maintaining a constant delayτ and varying the cyclical frequency α. The wavelength position of atunable laser (e.g., the tunable laser 110) may be kept constant, whilethe frequency of the ADC circuit may be varied to process the completesignal. The frequency may be changed by use of a frequency shifter(e.g., the frequency shifter 150 in FIG. 1). As an example, the initialconfiguration may acquire a first processed channel 702. Afteracquisition of the first processed channel 702 is complete, thefrequency of the ADC may be varied to acquire a second processed channel704 or a third processed channel 706. This may be used when thebandwidth of the ADC circuit is not sufficiently large (e.g., asillustrated in FIG. 2B) to capture the desired spectral range.

FIG. 7B shows a CAF table 700B generated by varying the delay τ andmaintaining a cyclical frequency α. The delay τ may be varied by tuningthe wavelength position of a tunable laser (e.g., the tunable laser110). A first processed channel 712 may be acquired with an initialwavelength position of a tunable laser. The wavelength position may thenbe varied to acquire a second processed channel 714, a third processedchannel 716, a fourth processed channel 718, a fifth processed channel720, and a sixth processed channel 722, etc. As discussed in relation toFIG. 6, the extent of the processed channels along the ordinate axis(Delay τ) is merely for purposes of illustration and it will beappreciated that, in practice, the processed channels correspond to aset of lines extending over a range of cycle frequencies. One ofordinary skill in the art would recognize many variations,modifications, and alternatives.

In a general case, a CAF table is completed by the combination of bothtiling strategies seen in FIG. 7A and FIG. 7B. Thus, in order togenerate a CAF table that extends over a range of cyclic frequencies anda range of delay values, the methods of FIG. 7A and FIG. 7B may be usedin combination. In practice, a CAF table can be generated byparallelizing the acquisition processes. To accomplish this, backplanecircuits are added after a wavelength-division demultiplexer (WDM DMUX).The hybrid photonic-electronic processors seen in embodiments above mayfurther accomplish this by replacing a single-wavelength carrier (e.g.,the fixed-frequency laser 120 in FIG. 1, etc.) with a comb source (CS)possessing multiple, mutually coherent carriers.

FIG. 8 is a simplified schematic diagram illustrating a hybridphotonic-electronic processor utilizing a parallelized acquisitionsystem according to an embodiment of the present invention. Theparallelized hybrid photonic-electronic processor 800 illustrated inFIG. 8 shares some common elements with the hybrid photonic-electronicprocessor 100 illustrated in FIG. 1 and the description provided inrelation to FIG. 1 is applicable to FIG. 8 as appropriate.

Backplane circuits (e.g., a 90-degree hybrid 824, the corresponding setof balanced receivers 826 and 827, and the ADC elements 828 and 829) areadded after a wavelength division demultiplexer (WDM DMUX) 870 such thatCAF maps can be generated. The processor scaling is accomplished byreplacing the single-wavelength carrier (e.g., the fixed-frequency laser120 of FIG. 1) by a comb source 820 providing multiple, mutuallycoherent carriers 821, 822, 823, through 831, representing mutuallycoherent carriers 1 through n.

Each comb carrier of the mutually coherent carriers 821 through 831 ismodulated via a similar optoelectronic modulator chain (e.g., the firstoptoelectronic modulator 806, the second optoelectronic modulator 808,and the third optoelectronic modulator 818) as discussed in previousembodiments and provided to a dedicated detector element 816 andbackplane circuits via the WDM DMUX 870. As illustrated in FIG. 8, eachof the frequency comb carriers (e.g., the first coherent carrier 821,the second coherent carrier 822, etc.), after modulation by firstoptoelectronic modulator 806, propagate through the first dispersiveline 812 and experience a fixed delay Δτ between adjacent carriers asdiscussed more fully in relation to FIG. 9.

In one implementation, a CAF table can be filled using the processillustrated in FIG. 7A. This may be accomplished by fixing thewavelength of the tunable laser 810 and varying the frequency of the setof ADC elements 828, 829, etc. by passing various coherent carriers(e.g., a first coherent carrier 821, a second coherent carrier 822,etc.) from the comb source 820. In another implementation, a CAF tablemay be filled using the process illustrated in FIG. 7B through tuning ofthe tunable laser 810. In alternative implementations, the CAF table iscompleted by using both acquisition methods shown in FIG. 7A and FIG.7B.

It should be noted that although second optoelectronic modulator 808,antenna 802, dispersive element 814, and coherent detector 816 producingboth quadrature and phase (I/Q) output in order to drive a thirdcoherent optoelectronic modulator 818 are utilized in FIG. 8, this isnot required and elements utilized to provide the delayed signal y(t−τ)can be utilized, for example, as illustrated in FIG. 1. Additionally,filter 857 is optional. As discussed in relation to FIG. 1, thefrequency shifter 850 may be configured to shift the frequenciesprovided by the comb source by an arbitrary amount.

FIG. 9 is a plot showing CAF tables generated using a parallelizedhybrid photonic-electronic processor. Frequency comb carriers (e.g., thefirst coherent carrier 821 in FIG. 8, etc.) experience a fixed delay Δτbetween adjacent carriers after passing through a first dispersiveelement (e.g., the first dispersive line 812). This is equivalent tosampling multiple delay states of CAFs and acquiring C_(α) ^(T)(τ_(k)),where k=1,2,3, . . . n, where n is the total number of frequency combcarriers. The dispersion D₁ introduced by the first dispersive element(e.g., the first dispersive line 812) and frequency comb pitch Δf (e.g.,the spacing between the coherent carriers of CS 820) define theresolution of the delay sampling period (e.g., the fixed delay Δτ). Anyother delay value is acquired by tuning a tunable laser (e.g., thetunable laser 810) to sweep the tunable laser over a range offrequencies corresponding to sweep time δτ 904. A complete coverage ofdelay ranges corresponds to Δτ=δt, imposing a direct relation amongstdispersion elements, tunable laser wavelength tuning range, and combcarrier pitch Δf.

In some embodiments, the CAF table can be generated rapidly by sweepingthe delay range in a continuous manner. In these embodiments, a tunablelaser, rather than being operated at a fixed wavelength, i.e., occupyinga fixed wavelength position, for the duration of the delay-productacquisition, is swept over a range of wavelengths as illustrated in FIG.10.

FIG. 10 is a plot showing the relation between the maximal delay range(τ_(RANGE)) of a tunable laser, the time utilized to sweep the laserwavelength over the delay range (t_(Sweep)), and the time to acquire asingle delay state (t_(Dwell)). During operation, the tunable laser iscontinuously tuned to provide delays over the maximal delay rangeτ_(RANGE). The instantaneous value of the delay τ(t) is defined by:

$\begin{matrix}{{\tau(t)} = {D \times L \times \frac{\delta\lambda}{\delta t} \times t}} & (7)\end{matrix}$

where D is the value of the dispersion measured in ps/km-nm, L is thelength of the dispersive line in km,

$\frac{\delta\lambda}{\delta t}$

is the rate of laser wavelength sweep in nm/sec, and t is the timemeasured in seconds.

FIG. 11 is a plot showing the delay variation during acquisition time.The plot shows delay τ increasing as the tunable laser is swept over arange of wavelengths. Thus, the CAF table is generated (e.g., at theoutput of the backend processor 830) in the vertical acquisitionscenario illustrated in FIG. 7B. If the dwell time used to acquire asingle delay state t_(Dwell) is small in comparison to the time utilizedto sweep the laser wavelength over the delay range t_(Sweep), then theslope of τ_(RANGE) to t_(Sweep) can be considered as negligible,resulting in a substantially constant value for the delay τ.Accordingly, the delay values in the set of delays (τ_(set)) is limitedto a small value centered on τ_(m), representing the delay correspondingto the measurement.

FIG. 12 is a plot illustrating constraints defined by the tunable lasersweep rate. As illustrated in FIG. 12, when the acquisition time isspecified (e.g., t_(Dwell) is constant) and the number of delay statesin the CAF table are given as N, the variation in the laser sweep ratealso mandates a change in the Δτ and t_(Sweep). The plot shows a firstsweep and second sweep with a constant t_(Dwell) and wheret_(Sweep1)<t_(Sweep2). At the k^(th) delay state in the first sweep, thechange in the delay over the sweep time t_(Sweep1) takes a first valueτ_(set1). At the k^(th) delay state in the second sweep, the change inthe delay over the sweep time t_(Sweep2) takes a second value τ_(set2).To maintain a constant t_(Dwell), τ_(set1) is greater than τ_(set2).

The variance in the delay as a function of dwell time t_(dwell) can bedefined for a given laser sweep rate and acquisition time:

$\begin{matrix}{\tau_{set} = {D \times L \times \frac{\delta\lambda}{\delta t} \times t_{Dwell}}} & (8)\end{matrix}$

where τ_(set) is the change in delay over the dwell time, D is thedispersion of a dispersive element, and L is the length of thedispersive element. For a given gain expected from the CAF table, thereis a uniquely defined t_(Dwell) value. As an example, for a sweep rateof δλ/δt=2000 nm/s, dispersion of D=120 ps/km-nm and dispersion elementlength L=2 km and dwell time of t_(Dwell)=10 μs, the change in delaytime during acquisition time is Δτ=48×10⁻⁸×10 μs=4.8 ps, mandating thatany signal variation during the acquisition time remains slower thanapproximately 5 picoseconds. In practice, this condition is easilysatisfied: signal bandwidth would need to exceed approximately 100 GHzto violate the limit imposed by the specific choice of parameters usedin this case. Ultimately, the application (e.g., the nature of thesignal inputs of the processors) will define an acceptable range of Δτ.

FIG. 13 is a plot illustrating two acquisition conditions in which thesweep time of the tunable laser is constant and the measurement periodsdiffer. In FIG. 13, the laser sweep rate is equal in both acquisitionconditions, i.e., t_(Sweep) is constant. For the first acquisitioncondition, the measurement time is set at t_(Dwell1). In this case,given a measurement time of t_(Dwell1), the change in the delay over thesweep time t_(Sweep1) is a first value τ_(set1). For the secondacquisition condition, the measurement time is decreased to t_(Dwell2).In this second case, given the reduced measurement time of t_(Dwell2),the change in the delay over the sweep time t_(Sweep1) is reduced to asecond value τ_(set2). Thus, shorter measurement times will producedecreased changes in delay during a measurement.

Thus, considering FIGS. 12 and 13, an increase in the sweep time willreduce the changes in delay during a measurement (FIG. 12) and adecrease in the measurement time will reduce the changes in delay duringa measurement (FIG. 13). One of ordinary skill in the art wouldrecognize many variations, modifications, and alternatives.

Physical parameters described above are interdependent and definemultiple operating regimes of the processor. An example of a regime isgiven. A controller may assist a user with operation of the processorthrough the use of controls. A user selects the desired range over whichthe delay time is to be varied: a control algorithm may convert thisparameter to tunable laser wavelength start and stop values, definingthe τ_(Range) parameter (e.g., the user chooses −15 ps to +15 ps; thecontrol algorithm may then instruct the laser being swept in wavelengthto sweep from 1500 nm-1580 nm). The user selects τ resolution: thecontrol may then calculate an amount of τ bins it has to cover (e.g.,300 τ bins for 0.1 ps resolution over a 30 ps range). The user selectsalpha resolution or processing gain: the controller may convert this todwell time t_(Dwell) and calculates how many samples N to collect duringt_(Dwell) (e.g., for a dwell time of 10 ms, 2500 samples at a samplingrate of 250 MHz can be utilized). The control algorithm calculates thetotal measurement time and the laser sweep speed: laser sweep speed

$\frac{\delta\lambda}{\delta t} = {\Delta{\tau/( {t_{Dwell} \times N} )}}$

(e.g., 80 nm/(10 ms×300)=26,666 nm/s is greater than a maximum lasersweep speed (˜2500 nm/s)). The controller may accommodate by applyingthe maximum laser sweep speed and maintaining t_(Dwell), supplying only300 points out of 4000 collected. Alternatively, the user is invited toselect a higher delay resolution, to integrate longer, or switch thechange in dispersive length.

FIG. 14 is a simplified flowchart illustrating a method of generating across-ambiguity function (CAF) table according to an embodiment of thepresent invention. The method 1400 includes a) generating a firstoptical carrier (1410) and b) modulating the first optical carrier witha first input signal of a set of input signals to provide a firstmodulated signal (1412). The first optical carrier can be a fixedwavelength optical carrier. Modulating the first optical carrier caninclude the use of a single-sideband modulator, a dual-port Mach-Zehndermodulator, or a dual-port, nested Mach-Zehnder device. The method alsoincludes c) generating a second optical carrier at a predeterminedfrequency f_(i) (1414), d) modulating the tunable optical carrier with asecond input signal of the set of input signals to provide a secondmodulated signal (1416), e) propagating the second modulated signalthrough a dispersive element to provide a dispersed signal (1418), andf) detecting the dispersed signal (1420).

The method further includes g) modulating the first modulated signalwith the dispersed signal to provide a product signal (1422), h)converting the product signal into an electrical signal (1424), i)digitizing the electrical signal to provide a digitized signal (1426),and j) performing a discrete Fourier transform on the digitized signal(1428). In some embodiments, the product signal is filtered prior toconverting the product signal into an electrical signal. In this case,the filtered product signal is converted into an electrical signal atstep 1424. A determination is made with respect to whether a desiredfrequency range between a first frequency f₁ and a second frequency f₂has been covered (1430). If not, then the predetermined frequency ismodified (1432) and steps c) through j) are performed in an iterativemanner until the range of frequencies f₁<f_(i)<f₂ has been covered. Oncethe range has been covered, then the method ends (1434).

In some embodiments, the method also includes k) converting the firstoptical carrier into a second electrical signal, l) digitizing thesecond electrical signal to provide a second digitized signal, and m)performing a discrete Fourier transform on the second digitized signal.The method can also include shifting the first optical carrier to ashifted frequency, performing b) through j) as discussed above, anditeratively performing c) through j) over the range of frequenciesf₁<f_(i)<f₂.

It should be appreciated that the specific steps illustrated in FIG. 14provide a particular method of generating a CAF table according to anembodiment of the present invention. Other sequences of steps may alsobe performed according to alternative embodiments. For example,alternative embodiments of the present invention may perform the stepsoutlined above in a different order. Moreover, the individual stepsillustrated in FIG. 14 may include multiple sub-steps that may beperformed in various sequences as appropriate to the individual step.Furthermore, additional steps may be added or removed depending on theparticular applications. One of ordinary skill in the art wouldrecognize many variations, modifications, and alternatives.

One of ordinary skill in the art will appreciate that othermodifications to the apparatuses and methods of the present disclosuremay be made for implementing various applications of the hybridphotonic-electronic processor without departing from the scope of thepresent disclosure.

The examples and embodiments described herein are for illustrativepurposes only. Various modifications or changes in light thereof will beapparent to persons skilled in the art. These are to be included withinthe spirit and purview of this application, and the scope of theappended claims which follow.

1. An apparatus comprising: a fixed-wavelength laser operable to providea fixed wavelength optical carrier; a first optical modulator opticallycoupled to the fixed-wavelength laser and configured to modulate thefixed wavelength optical carrier with a first input signal of a set ofinput signals; a tunable laser operable to provide a tunable opticalcarrier; a second optical modulator optically coupled to the tunablelaser and configured to modulate the tunable optical carrier with asecond input signal of the set of input signals; a dispersive elementcoupled to the second optical modulator; a first optical detectorcoupled to the dispersive element; a third optical modulator opticallycoupled to the first optical detector and the first optical modulator;an optical 90-degree hybrid element optically coupled to the thirdoptical modulator; and a plurality of optical detectors opticallycoupled to the optical 90-degree hybrid element and configured toconvert outputs of the optical 90-degree hybrid element into electricalsignals.
 2. The apparatus of claim 1 further comprising an opticalfilter optically coupled to the third optical modulator and disposedbetween the third optical modulator and the optical 90-degree hybridelement and configured to integrate an output of the third opticalmodulator.
 3. The apparatus of claim 1 wherein the first opticalmodulator comprises a single-sideband modulator.
 4. (canceled)
 5. Theapparatus of claim 1 wherein the first optical modulator furthercomprises a dual-port, nested Mach-Zehnder device configured to modulatethe phase and amplitude of the first input signal of the set of inputsignals onto the fixed wavelength optical carrier.
 6. The apparatus ofclaim 1 wherein the first optical detector comprises a coherent detectorconfigured to receive a modulated signal.
 7. The apparatus of claim 1wherein the first optical detector comprises an incoherent opticaldetector configured to receive a signal modulated onto the tunableoptical carrier.
 8. The apparatus of claim 1 further comprising a seconddispersive element optically coupled to an output of the first opticalmodulator and an input of the third optical modulator and configured todelay the first input signal of the set of input signals modulated ontothe fixed wavelength optical carrier.
 9. The apparatus of claim 1wherein the fixed-wavelength laser comprises an optical frequency combproviding a plurality of fixed wavelength optical carriers.
 10. Anapparatus comprising: an optical source operable to provide a frequencycomb including a plurality of optical carriers; a first opticalmodulator optically coupled to the optical source and electricallycoupled to a first antenna operable to receive a first input signal of aset of input signals; a tunable laser operable to provide a tunableoptical carrier; a second optical modulator optically coupled to thetunable laser and electrically coupled to a second antenna operable toreceive a second input signal of the set of input signals; a dispersiveelement optically coupled to the second optical modulator; a firstoptical detector optically coupled to the dispersive element; a thirdoptical modulator optically coupled to the first optical modulator andelectrically coupled to the first optical detector; awavelength-division demultiplexer optically coupled to the third opticalmodulator and the optical source; a plurality of optical 90-degreehybrid elements, each of the plurality of optical 90-degree hybridelements being coupled to the wavelength-division demultiplexer; aplurality of optical detectors, each of the plurality of opticaldetectors being coupled to an output of one of the plurality of optical90-degree hybrid elements; and a plurality of analog-to-digitalconverters, each of the plurality of analog-to-digital converters beingelectrically coupled to one of the plurality of optical detectors. 11.(canceled)
 12. The apparatus of claim 10 further comprising a seconddispersive element optically coupled to the first optical modulator anddisposed between the first optical modulator and the third opticalmodulator.
 13. The apparatus of claim 10 further comprising an opticalfilter optically coupled to the third optical modulator and disposedbetween the third optical modulator and the wavelength-divisiondemultiplexer.
 14. (canceled)
 15. The apparatus of claim 10 wherein thefirst optical detector comprises a coherent detector configured toreceive a modulated signal.
 16. The apparatus of claim 10 wherein thefirst optical detector comprises an incoherent optical detectorconfigured to receive a signal modulated onto the tunable opticalcarrier.
 17. The apparatus of claim 10 further comprising a seconddispersive element optically coupled to an output of the first opticalmodulator and an input of the third optical modulator and configured todelay the first input signal of the set of input signals modulated ontoa fixed wavelength optical carrier.
 18. A method of generating across-ambiguity function (CAF) table, the method comprising: a)generating a first optical carrier; b) modulating the first opticalcarrier with a first input signal of a set of input signals to provide afirst modulated signal; c) generating a second optical carrier at apredetermined frequency f_(i); d) modulating the second optical carrierwith a second input signal of the set of input signals to provide asecond modulated signal; e) propagating the second modulated signalthrough a dispersive element to provide a dispersed signal; f) detectingthe dispersed signal; g) modulating the first modulated signal using thedispersed signal to provide a product signal; h) converting the productsignal into an electrical signal; i) digitizing the electrical signal toprovide a digitized signal; j) performing a discrete Fourier transformon the digitized signal; and iteratively performing c) through j) over arange of frequencies f₁<f_(i)<f₂.
 19. The method of claim 18, whereinthe first optical carrier comprises a fixed wavelength optical carrier.20. The method of claim 18, further comprising filtering the productsignal prior to converting the product signal into the electricalsignal.
 21. The method of claim 18, wherein modulating the first opticalcarrier comprises using a single-sideband modulator.
 22. The method ofclaim 18, further comprising: k) converting the first optical carrierinto a second electrical signal; l) digitizing the second electricalsignal to provide a second digitized signal; and m) performing adiscrete Fourier transform on the second digitized signal; shifting thefirst optical carrier to a shifted frequency; performing b) through j);and iteratively performing c) through j) over the range of frequenciesf₁<f_(i)<f₂.
 23. The method of claim 18, wherein converting the productsignal into the electrical signal comprises: providing the productsignal to an optical 90-degree hybrid element optically coupled to anoptical modulator; and converting outputs of the optical 90-degreehybrid element into the electrical signal using a plurality of opticaldetectors optically coupled to the optical 90-degree hybrid element.